Jaap's Puzzle Page

Cohan Circle / Arusloky

The Cohan Circle consists of two overlapping circular discs. The distance
between the centres is exactly one radius length. The discs are made up of
many pieces with curved sides, which allow them to be rotated any number of
1/6th turns. Each disc has 6 blank triangles, and 12 petal-shaped pieces. As
some pieces are shared between the two discs, there are all together 10
triangles and 19 petals.

The 19 petals come in 4 colours - 6 red, 6 green, 6 blue, and 1 white.
The aim is to make the 6 peripheral pieces of each disc of a single
colour, forming two circles, and also place the white piece in the
centre.

This puzzle was invented and patented by Hooshang Cohan, number US 4,580,783
published 8 April 1986. It may have been sold as the 'Magic Circle Puzzle',
as the solution booklet has that title covered by a label with the correct
name.

Arusloky is a recent version of this puzzle, made in Spain. It has a
slightly different colour scheme, with 6 red petals, 6 yellow petals,
and everything else blue. Its starting configuration has a red and a
yellow circle.

If your browser supports it, you can click on the link below to play with a
Javascript version of both the Cohan Circle and Arusloky.

Links to other useful pages:

The number of positions:

There are 19 coloured pieces, so there are at most 19! positions. This limit
is not reached because The Cohan Circle has three sets of six indistinguishable pieces.
This leaves only 19!/6!3 = 325,909,584 positions. If you consider
the three coloured sets to be equivalent, then you can divide by a further 3!
to get 54,318,264 positions.
Arusloky has two sets of six and one set of seven indistinguishable pieces.
It therefore has 19! / (6!2·7!) = 46,558,512 positions.

I have used a computer to calculate God's algorithm for the Cohan Circle. This first table shows the
results if the two circles may be any colour. Any position can be solved in at most 14 moves, or 20 if
every 1/6 of a turn is counted as a separate move.

Face turn metric

Sixth

turn

metric

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Total

0

6

6

1

24

24

2

24

48

72

3

12

96

96

204

4

96

288

192

576

5

48

432

768

360

1,608

6

12

384

1,536

1,920

672

4,524

7

216

1,920

4,800

4,512

1,344

12,792

8

72

1,656

7,656

13,776

10,368

2,688

36,216

9

6

888

8,472

26,832

37,416

23,292

5,220

102,126

10

288

6,744

37,056

85,908

96,984

51,768

9,120

287,868

11

48

3,816

38,160

141,492

260,832

248,160

105,924

11,448

809,880

12

1,392

28,188

177,636

506,928

778,632

607,140

159,156

4,800

2,263,872

13

264

15,624

168,576

759,792

1,800,432

2,286,288

1,119,540

86,400

120

6,237,036

14

48

5,904

122,160

887,700

3,250,272

6,454,830

5,176,068

709,416

2,496

16,608,894

15

1,560

63,696

789,708

4,544,274

14,003,772

17,464,728

3,783,360

25,164

40,676,262

16

264

22,056

471,384

4,445,148

21,498,660

41,519,520

13,868,748

160,134

81,985,914

17

24

3,672

149,352

2,358,984

18,030,234

56,024,250

30,471,336

609,204

24

107,647,080

18

168

15,528

430,644

5,324,604

27,410,358

26,806,164

1,134,564

48

61,122,078

19

264

12,372

281,952

2,435,406

4,791,876

509,670

24

8,031,564

20

564

12,024

52,134

16,266

80,988

Total

6

60

300

1,494

7,296

35,472

172,572

834,492

3,964,452

17,925,906

68,603,088

151,332,498

80,574,234

2,457,618

96

325,909,584

This table shows the results if the circles must have particular colours. Any position can be solved in
at most 17 moves, or 23 if every 1/6 of a turn is counted as a separate move.

Here are the results for the Arusloky. This first table shows the results if the two circle colours
may be swapped so there are two solutions. Any position can be solved in at most 14 moves, or 20 if
every 1/6 of a turn is counted as a separate move.

Face turn metric

Sixth

turn

metric

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Total

0:

2

2

1:

8

8

2:

8

16

24

3:

4

32

32

68

4:

32

96

64

192

5:

16

144

256

120

536

6:

4

120

512

636

224

1,496

7:

72

608

1,576

1,488

448

4,192

8:

16

496

2,348

4,520

3,472

840

11,692

9:

2

232

2,544

8,464

12,300

7,668

1,572

32,782

10:

56

1,896

11,220

27,524

32,160

16,136

2,376

91,368

11:

8

888

10,716

44,052

85,032

80,864

29,260

2,240

253,060

12:

288

7,432

52,244

162,592

256,720

177,912

33,772

592

691,552

13:

48

3,824

46,584

235,762

591,050

694,446

249,480

11,392

8

1,832,594

14:

1,144

29,972

253,296

1,023,098

1,953,822

1,173,934

93,470

160

4,528,896

15:

208

11,748

184,548

1,245,360

3,890,792

3,779,374

484,264

1,792

9,598,086

16:

24

2,356

72,648

875,036

4,595,956

7,487,246

1,607,840

9,546

14,650,652

17:

176

11,992

253,348

2,253,812

6,564,972

2,629,030

31,300

11,744,630

18:

8

440

17,044

265,496

1,447,556

1,237,330

35,884

3,003,758

19:

120

3,052

33,622

69,858

6,158

6

112,816

20:

8

28

56

16

108

Total:

2

20

100

482

2,232

10,344

49,264

230,884

1,046,978

4,360,348

13,866,932

20,772,224

6,133,832

84,864

6

46,558,512

And finally, this table shows the results if the Arusloky has only one solved position. This can
be reached from any position in at most 15 moves, or 22 if every 1/6 of a turn is counted as a separate move.

Face turn metric

Sixth

turn

metric

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Total

0:

1

1

1:

4

4

2:

4

8

12

3:

2

16

16

34

4:

16

48

32

96

5:

8

72

128

60

268

6:

2

60

256

318

112

748

7:

36

304

788

744

224

2,096

8:

8

248

1,174

2,260

1,736

420

5,846

9:

1

116

1,272

4,232

6,150

3,834

786

16,391

10:

28

948

5,610

13,758

16,072

8,064

1,208

45,688

11:

4

444

5,358

22,022

42,442

40,258

14,750

1,348

126,626

12:

144

3,716

26,110

80,992

127,024

88,646

19,282

728

346,642

13:

24

1,912

23,264

117,171

289,206

338,453

140,389

13,072

196

923,687

14:

572

14,974

125,532

495,119

927,789

650,840

111,670

3,338

16

2,329,850

15:

104

5,878

91,974

603,414

1,831,240

2,107,752

614,308

29,032

216

5,283,918

16:

12

1,218

37,564

448,179

2,324,322

4,591,719

2,299,538

180,077

1,962

4

9,884,595

17:

104

7,200

162,248

1,522,560

5,615,744

5,304,343

741,553

11,672

32

13,365,456

18:

4

512

20,669

382,822

2,791,664

5,621,094

1,675,960

47,070

190

10,539,985

19:

8

694

23,010

352,898

1,660,611

1,283,154

92,716

762

3,413,853

20:

4

116

4,714

61,760

158,713

43,429

964

269,700

21:

2

102

1,057

1,626

224

3,011

22:

2

3

5

Total:

1

10

50

241

1,116

5,172

24,632

115,442

523,721

2,195,665

7,454,916

16,276,352

15,687,226

4,073,080

198,709

2,179

46,558,512

Notation:

Let a clockwise 60 degree rotation of the left disc be denoted by L.
Rotations of 120, 180, 240, 300 degrees are then denoted by L2, L3, L4
and L5. Note that L5 can also be considered an anti-clockwise 60 degree
turn, and is therefore also denoted by L'. Turns of the right disc are
denoted in the same way, but using the letter R.

Terminology:

The left circle is the 6 piece locations that lie on the rim of the left
disk. Eventually all the pieces of one colour will be placed in the left circle.
Similarly, the right circle is the 6 locations on the rim of the
right disk, which will be made another colour.

The left extended circle is the left circle plus the horizontal
piece location at the right hand side. Note that the 7 pieces in the left extended
circle will remain in the left extended circle if you do R2, R4, or L moves.
Similarly the right extended circle is the right circle plus the left
horizontal piece location.

The spokes of the left disk are the six locations of the left disk that
meet in the centre, i.e. those locations that lie inside the left circle. Similarly
the spokes of the right disk are the six locations that meet in the centre
of the right disk.

The middle is the central horizontal location where the white piece
should be when the puzzle is solved. Note that this is a left spoke as well as
a right spoke.

Solution:

Phase 1: Put the green/yellow pieces in the left extended circle.

Try to put as many green (Cohan Circle) or yellow (Arusloky) pieces in the left Ŕxtended circle as you can before doing the steps below.

Find any green/yellow piece that is not yet in the left extended circle.

By doing one of the following steps, move the green/yellow piece so that it becomes the top right spoke of the right disk:
1. If it is one of the left spokes, turn the left disk so that the piece is in the middle, and then do R2.
2. If it is the bottom right spoke of the right disk, then do R4.
3. If it lies in the right circle, then do R2 or R4 to bring the piece into the left disk, and do step 1.

Turn the left disc so that a non-green/non-yellow piece lies at the top right of the left circle (the top left spoke of the right disk).

Do R3 L2 R L4 R' L2 R to insert the green/yellow piece into the left extended circle.

Repeat b-e for the remaining green/yellow pieces.

The left extended circle now contains six green/yellow pieces and one of another colour. If this seventh piece is red, then
use steps b-e to place any blue in the left extended circle in place of the red one.

Phase 2: Put the red pieces in their extended circle.

Turn the puzzle around, so that the green/yellow pieces are now in the right extended circle. In the steps below, the red pieces will be placed in the left extended circle in much the same way as the green/yellow ones were in phase 1.

Try to put as many red pieces in the left circle as you can without disturbing the green/yellow pieces from the right extended circle. Note that if you use only L2, L4, and moves of the right disk then the green/yellows will be safe.

Find any red piece that is not yet in the left extended circle.

By doing one of the following steps, move the red piece so that it becomes the top right spoke of the right disk:
If the red piece is one of the left spokes, turn the left disk so that the piece is in the middle, and then do R2.
If the red piece is the bottom right spoke of the right disk, then do R4 so that it becomes the top right spoke.

If necessary do L2 or L4 so that a non-red piece lies at the top right or bottom right of the left circle (the top-left/bottom-left spoke of the right disk).

If a non-red piece lies at the top right of the left circle, then do R3 L2 R L4 R' L2 R to insert the red piece there.

If a non-red piece lies at the bottom right of the left circle, then do L2 R2 L4 followed by R3 L4 R' L2 R L4 R' to insert the red piece.
Note that the first three moves put the red piece at the bottom right spoke of the right disk, and the rest of the sequence is the mirror image of the sequence of step f.

Repeat c-g for the remaining red pieces.

Phase 3: Make the circles.

Find the non-red piece in the left extended circle. If it is not one of the spokes of the right disk, then do L2 or L4 to make it so.

If necessary, do R2 or R4 so that the left circle is completely red.

Find the non-green piece in the right extended circle.

If it is at a left spoke, then a L2 or L4 turn will complete the green circle, otherwise do one of the following, depending on where in the right circle it lies:
Top: R2 L4 R2 L2 R4
Top right: R2 L2 R2 L4 R4
Bottom right: R4 L4 R4 L2 R2
Bottom: R4 L2 R4 L4 R2

Phase 4: Place the white petal in the centre.

If the white piece is in the left disk, then turn the puzzle around so that it lies in the right disk.